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Tautological relations and the \(r\)-spin Witten conjecture. (Relations tautologiques et la conjecture de Witten sur l’espace des structures \(r\)-spin.) (English. French summary) Zbl 1203.53090

The authors give a simple proof of the fact that the group action, introduced by A. Givental on Gromov-Witten potentials respect tautological relation in the cohomology ring of the moduli space \(\overline{\mathcal{M}}_{g,m}\) of stable pointed maps [Y. P. Lee, C. Teleman].
The authors observe that in any semi-simple GW theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the geometric GW potential coincides with the potential constructed via Givental’s group action. They also show that their results suffice to deduce the statement of a Witten conjecture relating the \(r\)-KdV hierarchy to the intersection theory on the space of \(r\)-spin structures on the stable curves.

MSC:

53D55 Deformation quantization, star products
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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